Let $\alpha>0$ and $\gamma>0$. We consider integral operator of the form$${\mathcal{G}}_{\phi_\gamma}f(x):=\frac{1}{\Psi_\gamma (x)}\int_0^x (1-\frac{y}{x})^{\alpha-1}\phi_\gamma(y) f(y)dy,\,\,\,\, x>0.$$This paper is devoted to the study of the infinity behavior of ${\mathcal{G}}_{\phi_\gamma}$. We also provide separately result on the similar problem in the weighted Lebesgue space.